| Résumé: | The volume cocycle of a Euclidean building was introduced by Bruno Klingler. Using a Poisson type transform, one may morally compute its norm, but the problem of determining its growth is an open problem, even in the $\widetilde{A}_2$ case. We give a complete answer in the case of a regular tree of finite valency, which was independently obtained in a stronger result of Gournay and Jolissaint, and a polynomial bound for a product of regular trees. |
| Résumé: | The class of highly-arc-transitive was defined by Cameron, Praeger and Wormald in an influential paper that appeared in 1993. In this talk, I will describe how graphs from this class regularly pop up in group theoretical work. I will also outline graph theoretical work to describe the structure of these graphs and resolve the many questions and conjectures made by Cameron, Praeger and Wormald. |
| Résumé: | The notion of complete reducibility was introduced by J. P. Serre in 1998. It generalises the notion of a completely reducible module in classical representation theory. After giving an introduction to complete reducibility for algebraic groups, we discuss its impact on questions about the subgroup structure of exceptional algebraic groups. This includes recent and ongoing work with A. Litterick on classifying the subgroups of exceptional algebraic groups that are not completely reducible. The techniques used are a mix of standard representation theory, non-abelian cohomology and computational group theory. |
| Résumé: | Les masures, introduites en 2008 par Gaussent et Rousseau sont des généralisations des immeubles de Bruhat-Tits. Elles permettent notamment d'étudier les groupes de Kac-Moody sur les corps locaux. Rousseau a donné en 2011 une axiomatique de ces espaces, inspirée de celle des immeubles de Bruhat-Tits. Dans cet exposé, on présentera une nouvelle axiomatique des masures que j'ai proposée récemment, plus simple et plus proche de celle des immeubles et on l'illustrera avec l'exemple de la masure associée à $\mathrm{SL}_2$ affine. |
| Résumé: | A natural question on group actions goes as follows. "Suppose that an isometric group action on a metric space admits fixed points for certain (small) subgroups. Under which conditions, can we 'upgrade' them to the existence of fixed points for a bigger subgroup, ultimately, for the whole group?" One well-known answer to this question is a Helly-type theorem. In 1999 and 2006, Shalom gave other answers for actions on Hilbert spaces; they were based on (BG) ("Bounded Generation"). I will present the first solution to a natural question from Shalom's work that asks whether we can upgrade fixed points (in an intrinsic way) without employing (BG). It, in particular, helps us to prove Kazhdan's property (T) for some groups. |
| Résumé: | Etant donné un groupe localement compact $G$, on va s'intéresser aux propriétés asymptotiques des suites de réseaux de $G$ dont le covolume tend vers l'infini. Notamment on va étudier l'asymptotique des nombres de Betti renormalisé par le covolume et on montra que ces suites admettent des limites qu'on peut identifier avec les nombres de Betti $\ell_2$ du groupe $G$, généralisant donc un résultat de ABBGNRS. Pour montrer cela on considerera les actions de $G$ sur les espaces quotients $G/\Gamma$ et en particulier l'action limite associée : l'action de $G$ sur un ultraproduit régularisé. |
| Résumé: | Some years ago, Lachlan advanced a theory on homogeneity in relational structures which imposed a natural "hierarchy of complexity" on the universe of homogeneous relational structure. This theory was reworked by Cherlin in the 1990's with a view to understanding finite permutation groups from a model theoretic point of view. One upshot of all this is that we know the existence of an infinite family of theorems describing the so-called "relational complexity" of all finite permutation groups. The problem is that, although we know these theorems exist, and even have a "form" for them, nonetheless we do not yet have the precise statement of any of them. However Cherlin has conjecture what (part of) the first of these theorems should say, and we will discuss this conjecture at some length. There has also been substantial progress on this conjecture due to Cherlin himself, to Wiscons, and to myself and various co-authors. In particular the recent results that I will describe are due to myself, Pablo Spiga, Francis Hunt and Francesca Dalla Volta. The talk has elements of model theory, combinatorics and finite permutation groups, and should be accessible to all. |
| Résumé: | This talk is on a work in progress. Let $H$ be a profinite subgroup of a topological group $G$ such that $H$ is open and maximal in $G$. By examining the permutation representations of $G$ and $H$ acting by left multiplication on the coset space $G/H$, we can deduce strong structural results about the profinite group $H$. |
| Résumé: | L'objectif principal de l'exposé sera d'introduire un formalisme combinatoire et élémentaire pour l'étude des complexes cubiques spéciaux (introduits par Haglund et Wise) et à leurs groupes fondamentaux. En guise d'application, je donnerai des caractérisations purement algébriques de certaines propriétés de courbure négative : hyperbolicités au sens de Gromov, relative, et acylindrique. Finalement, je tâcherai de montrer dans le reste de l'exposé que ce formalisme s'adapte très bien à l'étude des groupes de tresses "graphées". |